Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,21)(20,23)(22,24)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(10,13)(11,16)(18,19)(20,24)(25,26,29,32)(27,33,35,30)(28,31,37,36)(34,39,40,38);(10,13)(14,15)(17,18,21,19)(20,23,24,22)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,11,12,16)(10,14,13,15)(17,19,21,18)(20,23,24,22)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,13)(15,16)(17,20,21,24)(18,22,19,23)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=fgh,d^2=f,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5084617);